Research Article Geometry Optimization Approaches of …downloads.hindawi.com/journals/js/2016/4869571.pdf · is usually very small. is paper is dealing with geometry optimization

Research ArticleGeometry Optimization Approaches of InductivelyCoupled Printed Spiral Coils for Remote Powering ofImplantable Biomedical Sensors

Sondos Mehri,1,2 Ahmed Chiheb Ammari,3,4

Jaleleddine Ben Hadj Slama,2 and Hatem Rmili3

1ENIT, University of Tunis El Manar, Belvedere, 1002 Tunis, Tunisia2SAGE, National Engineering School of Sousse, ENISo, University of Sousse, 4054 Sousse, Tunisia3Renewable Energy Research Group, Department of Electrical & Computer Engineering, Faculty of Engineering,King Abdulaziz University, Jeddah 21589, Saudi Arabia4MMA Laboratory, National Institute of the Applied Sciences and Technology, Carthage University, 1080 Tunis, Tunisia

Correspondence should be addressed to Ahmed Chiheb Ammari; [email protected]

Received 27 November 2015; Revised 22 February 2016; Accepted 6 March 2016

Academic Editor: Alberto J. Palma

Copyright © 2016 Sondos Mehri et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Electronic biomedical implantable sensors need power to perform. Among the main reported approaches, inductive link is themost commonly used method for remote powering of such devices. Power efficiency is the most important characteristic to beconsidered when designing inductive links to transfer energy to implantable biomedical sensors. The maximum power efficiencyis obtained for maximum coupling and quality factors of the coils and is generally limited as the coupling between the inductorsis usually very small. This paper is dealing with geometry optimization of inductively coupled printed spiral coils for powering agiven implantable sensor system. For this aim, Iterative Procedure (IP) and Genetic Algorithm (GA) analytic based optimizationapproaches are proposed. Both of these approaches implement simple mathematical models that approximate the coil parametersand the link efficiency values. Using numerical simulations based on Finite Element Method (FEM) and with experimentalvalidation, the proposed analytic approaches are shown to have improved accurate performance results in comparison with theobtained performance of a reference design case. The analytical GA and IP optimization methods are also compared to a purelyFinite ElementMethod based on numerical optimization approach (GA-FEM). Numerical and experimental validations confirmedthe accuracy and the effectiveness of the analytical optimization approaches to design the optimal coil geometries for the best valuesof efficiency.

1. Introduction

The treatment of a growing number of ailments has becomereal thanks to the revolution in semiconductor technologyand the steady growth of integrated circuit (IC) design forimplantable biomedical systems. An implantable electronicbiomedical sensor device is generally composed of an exter-nal controller and an implantable unit [1]. The external con-troller sends command to the implantable unit and processesreceived data from the implants through an interface unitwhich needs powering to perform its role. Among the mainreported approaches, inductive links are the most commonlyused methods for remote powering of such devices. This

method does not require any transcutaneous wires, whichmay cause infections, or any implantable batteries which needto be replaced at the end of their lifetime.Thepower efficiencyis the most important characteristic to be considered whendesigning inductive links to transfer energy to implantablesensing devices [2]. Maximum power efficiency is obtainedfor maximum coupling factors of the coils. This parameteris generally limited since the coupling between inductorsis usually very small. Improving power efficiency of aninductive link is crucial and essential to minimize the size ofthe external energy source, the electromagnetic heating of thetissue, the interference with other devices, and imperativelythe safety of the patient. Therefore, attention must be paid

Hindawi Publishing CorporationJournal of SensorsVolume 2016, Article ID 4869571, 11 pageshttp://dx.doi.org/10.1155/2016/4869571

2 Journal of Sensors

to the link design in order to obtain high transferred powerefficiency [2]. This includes proper optimization of coilgeometries [3], appropriate design of primary coil drivers[4], procedures to operate the link as close as possible toits theoretical efficiency limit [5], and also servo loops thatmay be used to automatically adjust the link parameters tochanging coupling and load conditions [6].

We are particularly concerned with the geometric opti-mization of inductively coupled printed spiral coils (PSC)for powering biomedical implantable sensor devices. Thisoptimization is a laboured assignment given the large num-ber of parameters characterizing the link performance. Theobjective is to optimize, within an acceptable design time,the PSC geometric parameters such that the maximum linkefficiency is guaranteed. This optimization problem is ofhigh dimensions, has high degree of nonlinearity, and isthus hard to solve. In this context, two main classes of opti-mization methods are generally used: analytic and numericapproaches. The analytic approaches are based on simplemathematical models that approximate both coil parametersand the link power efficiency. They are implemented usingeither iterative design procedures or advanced global searchmetaheuristics. The Iterative Procedure is a traditional pop-ular approach for optimizing the link coil geometries [7, 8].Advanced population based search strategies implementingGenetic Algorithms [9, 10] or particle swarm optimizers[11] can also be used for inductive link optimizations. Suchglobal approaches aim to efficiently explore the search spacesuch that near-optimal solutions are obtained and rapidconvergence is ensured [12]. For the numeric approaches,the optimization is also implemented through conventionalglobal search techniques such as Genetic Algorithms [13]or particle swarm optimizers [14]. The difference is that,in this case, the objective function is evaluated by numer-ical simulation using specific finite element software pack-ages. Numeric optimization methods have been previouslyimplemented for many RFID [15] and antenna [16] designcases.

This work aims to implement appropriate optimizationapproaches capable of achieving optimal link designs withinacceptable design times. Analytical approaches implement-ing either Iterative Procedures (IP) or Genetic Algorithms(GA) are first proposed. These approaches were originallypresented in [17]. This paper starts by reviewing the IPapproach and details the implementation of its differentsteps. The GA optimization for the chosen implant isthen developed and appropriate selection, crossover, andmutation operators are carefully designed to ensure rapidconvergence of the algorithm. Moreover, a purely numericalsimulation based optimization, GA-FEM, is developed. GA-FEM consists in coupling finite element numeric simulationmethod (FEM) with GA based search strategy [16, 18].The outperformance of GA-FEM is demonstrated. Com-paratively, efficiency degradations of proposed IP and GAdesigns are evaluated. Comparisons with a previous designstudied for the same implant in [19] are also performed.At the end, experimental measurements are implemented tovalidate the different design approaches. Reasonable errorsare observed for the analytical results despite the approximate

Internal coilExternal coil

V Skin

L1 L2

Cp1Cp2

C2

C1

R1 R2

+

−

M

Rload

Figure 1: Simplified model of the inductive power link.

mathematical models being used. Comparing IP to GA, theaccuracy and the effectiveness of GA are confirmed.

The remainder of the paper is as follows. The nextsection explains the basic theoretical background of inductivelink modeling and efficiency calculations. For an induc-tive link optimal geometric design, the Iterative Procedure(IP) together with the GA based analytical optimizationapproaches is presented in Section 3. The GA-FEM numericoptimization approach is described in Section 4. Section 5presents the experimental validationwith a comparative anal-ysis between the different design approaches. Conclusions aregiven in the last section.

2. Theoretical Background

This section first introduces the circuit model of an inductivelink, and then simple mathematical models that approximatecoil parameters and transfer power efficiency are presented. Abasic assumption that the secondary coil of the link is embed-ded in the human body is ignored since, referring to [20–22],it is shown that losses through biological tissues and liquidsare negligible at frequencies between several hundred kHzto 20MHz. For the study, 13.56MHz operating frequencyis selected. Thus, the surrounding media of the coils areneglected for the following coil modeling, simulation, andexperimentation.

2.1. Inductive Link Modeling. The behavior of an inductivelink comprising a primary circuit and a secondary circuit canbe treated at two different levels of abstraction: a descriptionof electromagnetic fields and a description of the circuitlevel. The latter description can be used to deduct thecharacteristics of the link. However, key circuit componentsuch as inductors, parasitic elements, and magnetic couplingcoefficients can be extracted using only the electromagneticfield description. Figure 1 shows a simplified diagram of aninductive power link. 𝐿

1and 𝐿

2are, respectively, the primary

and secondary coils. The primary coil is fixed outside thebody, whereas the secondary coil is attached to electronicsof the implant that are embedded in the human body.Coil windings are characterized by their respective parasiticresistance (𝑅

1, 𝑅2) and capacitance (𝐶

𝑝1, 𝐶𝑝2).

To increase the transfer efficiency, both sides of the linkare tuned to the same resonant frequency. This can be imple-mented by adding two capacitors 𝐶

1and 𝐶

2, respectively,

to the primary and secondary circuits to form primary andsecondary LC resonant circuits. There are four doubly tuned

Journal of Sensors 3

s

w di

do

Figure 2: Geometry of a square spiral coil.

link choices involving all possible combinations of currentand voltage input and output [4]. For example, a series-resonant has a current-source output characteristic and avoltage-source output is achieved through parallel resonanceof the secondary circuit. In Figure 1, the primary circuit usesa serial LC circuit to provide low impedance to the drivingside. The secondary circuit is tuned in parallel with betterdrive nonlinear rectifier loads. A parallel-tuned setup hasthe additional advantage that the parasitic coil capacitor isabsorbed in the resonance capacitor 𝐶

2, allowing for higher

transfer frequencies and thus smaller coils [4].

2.2. Planner Spiral Coil Quality Factor andMutual Inductance.Applying a magnetic field at the primary coil will induce acurrent flowing in the secondary circuit. The value of theinduced current is related to 𝐿

1and 𝐿

2inductances of both

primary and secondary coils. For the envisaged applications,a square spiral coil form is selected given that, for a definedboundary box, square spiral coils enable themaximumpowertransfer area [8]. For the chosen square spiral planar coil(Figure 2), the inductance is calculated according to thefollowing formula [19]:

𝐿 =

1.27 ⋅ 𝜇0⋅ 𝑛2𝑑avg

2[ln(

2.07

𝜑) + 0.18𝜑 + 0.13𝜑

2] , (1)

where 𝑛 is number of turns

𝑑avg =(𝑑𝑖+ 𝑑𝑜)

2;

𝜑 =(𝑑𝑜− 𝑑𝑖)

(𝑑𝑜+ 𝑑𝑖)

(2)

with 𝑑𝑜and 𝑑

𝑖being the outer and inner diameter, respec-

tively.

(i) Quality Factor. The inductor quality factor is an importantparameter that affects the link power efficiency. It is related tothe parasitic resistance and capacitance of the inductor. Tak-ing into account the skin effect, the total parasitic resistancecan be calculated as follows [8]:

𝑅𝑠= 𝑅dc

𝑡𝑐

𝛿 ⋅ (1 − 𝑒−𝑡𝑐/𝛿)(3)

with 𝛿 being the skin metal depth and 𝑅dc the DC resistance,expressed as follows:

𝑅dc = 𝜌𝑐

𝑙𝑐

𝑤 ⋅ 𝑡𝑐

;

𝛿 = √𝜌𝑐

𝜋 ⋅ 𝜇 ⋅ 𝑓; 𝜇 = 𝜇

𝑟⋅ 𝜇0,

(4)

where 𝑙𝑐is total length of the conductor, 𝑡

𝑐is conductor

thickness, 𝑤 is conductor width, 𝜌𝑐is metal resistivity, 𝜇 is

permeability constant, and 𝜇𝑟is relative permeability of the

conductor.Ignoring the parasitic capacitance of the circuit, the

quality factor of the Printed Circuit Coil (PSC) can beexpressed as 𝑄 = 𝜔𝐿/𝑅

𝑠[8].

(ii) Coils Mutual Inductance. The mutual inductance is a keyparameter for inductive power transfer. Assuming perfectalignment, the total mutual inductance between the two coilsis expressed by

𝑀 = 𝑔

𝑛1

∑

𝑖=1

𝑛2

∑

𝑗=1

(𝑀𝑖𝑗, 𝑟𝑖, 𝑟𝑗, 𝑑12) , (5)

where 𝑔 is factor depending on the shape of the coil (𝑔 = 1.3

for square shape), 𝑛1is number of turns of the primary coil,

𝑛2is number of turns of the secondary coil, 𝑟

𝑖is radius of 𝑖th

turn in primary coil, 𝑟𝑗is radius of 𝑗th turn in secondary coil,

and 𝑑12is distance between coils.

2.3. Inductive Link Power Efficiency. The highest voltage gainand efficiency across an inductive link (Figure 1) can beachieved when both LC tanks are tuned at the link operatingfrequency. Assuming 𝜔 = 𝜔

01= 𝜔02, the efficiency of

the transmitted power from source to load is dominated byprimary and secondary side efficiencies, 𝜂

1and 𝜂

2, and can

be expressed by

𝜂12

= 𝜂1⋅ 𝜂2

=𝑋 (𝑅2/𝑅load)

(1 + 𝑅2/𝑅load + 𝑋 (𝑅

2/𝑅load)) (1 + 𝑅

2/𝑅load)

,

(6)

where

𝑋 =𝑀2⋅ 𝜔2

𝑅1⋅ 𝑅2

= 𝑘2⋅ 𝑄1⋅ 𝑄2, with 𝑘 =

𝑀

√𝐿1⋅ 𝐿2

, (7)

where 𝑘 is the coupling coefficient and 𝑄1and 𝑄

2are the

quality factors of the unloaded primary and secondary coils,respectively.

By taking 𝑅load as optimum, the maximum efficiency canbe calculated as follows:

𝜂12

=𝑘2𝑄1𝑄2

(1 + √1 + 𝑘2𝑄1𝑄2)

2. (8)

For more details about how these formulas are obtained,one can consult [19, 21, 22].


Validation

(2) Applying of initial values

(1) Applying design constraints

No

Yes

(4) Optimizing secondary coil (𝜑2, w2)

(5) Optimizing primary coil (do1, w1)

(do, di, d12)

(s, w, 𝜑1, 𝜑2)

(3) Optimizing primary coil (do1, 𝜑1)

(6) Efficiency improvement < 0.1%

Figure 3: Iterative design procedure flowchart.

3. Analytic Design IP and GABased Approaches

This section presents Iterative Procedure (IP) and GeneticAlgorithm (GA) analytical approaches to optimally designthe coil geometries of two inductively coupled printed spi-ral coils for powering a particular biomedical implantablesystem. The chosen biomedical implantable sensor is usedfor a laboratory mouse and aims for biomedical and geneticresearches to investigate new treatments as presented in[19]. For such design approaches, some parameters areapplication constrained and others have to be carefullychosen to guarantee the best efficiency values of thelink.

3.1. The Iterative Procedure (IP). The implemented IterativeProcedure consists of six steps as presented in the flowchartof Figure 3.

Each step in this figure is explained and detailed asfollows.

(1) Applying Design Constraints. There is a set of constrainedparameters that are imposed by factors associated with thechosen implant system. These parameters usually define thesize constraints as limited by where the implant is located.Other factors are related to the fabrication technology. Thisconsists in theminimum size features that result in acceptablemanufacturing. For our case, the constrained parameters andtheir values are listed in Table 1.

(2) Initial Values. For this study, the primary coil outerdiameter 𝑑

𝑜1, the spacing between the conductors (𝑠

1, 𝑠2),

and the width of conductors (𝑤1, 𝑤2) for the primary and

secondary coils are the selected variables to be optimized.A set of initial values for these variables is defined before

Table 1: Design parameters limited by implant and fabricationtechnology.

Parameters ValueSecondary coil outer diameter (𝑑

𝑜2) 20 (mm)

Distance between coils (𝑑12) 30 (mm)

Minimum coils inner diameters (𝑑𝑖1and 𝑑

𝑖2) 5 (mm)

Minimum width of conductor∗ (𝑤) 150 (𝜇m)Minimum spacing between conductors∗ (𝑠) 150 (𝜇m)Operating frequency (𝑓) 13.56 (MHz)Substrate thickness (𝑡

𝑠) 1.5 (mm)

Conductor thickness (𝑡𝑐) 38 (𝜇m)

∗The conductor is copper on FR4 printed circuit board.

05

10152025303540

20 40 60 79

Effici

ency

(%)

99 119do1 (mm)

Figure 4: Calculated efficiency (𝜂12) versus (𝑑

𝑜1) for Step 3 opti-

mization.

starting the iterative optimization process. As shown in [8],a good choice for 𝑑

𝑜1initial value would be

𝑑𝑜1

= 𝑑12

∗ √2 = 42mm. (9)

The initial values selected for the other parameters are set totheir minimum values as follows: 𝑠

1= 𝑠2

= 0.15mm and𝑤1= 𝑤2= 0.15mm.

(3) Optimizing the Size and Fill Factor of Primary PSC(𝑑𝑜1, 𝜑1). The third step of the Iterative Procedure aims

to optimize the primary side efficiency. Plugging the coilparameters initial values defined in previous steps into (1) to(8), the link efficiency 𝜂

12is calculated for different 𝑑

𝑜1and

𝜑1(𝜑1

= (𝑑𝑜1

− 𝑑𝑖1)/(𝑑𝑜1

+ 𝑑𝑖1)) varying in a wide range

around their initial values. Since 𝑠1and 𝑤

1are kept constant

for this step, modifying 𝑑𝑜1and 𝜑

1directly affects 𝑘, 𝑄

1, and

𝜂12

as given by (8). It is obtained that 𝜂12

efficiency results,calculated for different 𝑑

𝑜1and 𝜑

1, converge to the highest

values for 𝜑1> 0.62. Selecting 𝜑

1= 0.62, 𝜂

12is computed

versus 𝑑𝑜1as reported in Figure 4.

Figure 4 clearly shows that themaximumefficiency valuesare obtained for 𝑑

𝑜1within 80 to 100mm range. For minimal

primary coil sizes, the best choice for 𝑑𝑜1

is set to 80mm.Setting 𝜑

1= 0.62, the inner diameter for the primary coil, 𝑑

𝑖1,

is calculated as 𝑑𝑖1= 10mm. For this step, almost 32% of total

efficiency is obtained. This efficiency value should improveonce𝑤

1and the secondary coils are optimized as shown next.


30

35

40

45

50

0.15 0.21 0.27 0.33 0.39 0.45

Effici

ency

(%)

w2 (mm)

Figure 5: Calculated efficiency (𝜂12) versus (𝑤

2) for Step 4 optimiza-

tion.

(4) Optimizing Fill Factor and Line Width of Secondary PSC(𝜑2, 𝑤2). The geometry of the primary PSC was temporarily

resolved in Step 3. In this step, we focus on the secondarycoil. Using the parameter values issued from previous steps,the link efficiency 𝜂

12is calculated while seeping 𝜑

2(𝜑2

=

(𝑑𝑜2

− 𝑑𝑖2)/(𝑑𝑜2

+ 𝑑𝑖2)) around its initial value and 𝑤

2set to

𝑤2min = 0.15mm. It is shown that high efficiency peaks are

obtained for 𝜑2> 0.65 values. However, 𝜑

2> 0.65 leads to

𝑑𝑖2

< 4mm which is less than 𝑑𝑖min. We choose 𝑑

𝑖2= 11mm

to leave more space for an electronic chip to be placed at thecenter of the secondary PSC.Thiswill reduce𝜑

2to 0.29. Using

this 𝜑2value, 𝜂

12is reported in Figure 5 for 𝑤

2varying from

𝑤2min = 0.15mm to 0.45mm. Figure 5 illustrates that the

efficiency increases almost linearly with increasing𝑤2values.

However, it is known that proximity effect losses increasewith increasing metal width [23]. To reduce proximity effectlosses without compromising too much efficiency, we select𝑤2

= 0.3mm which is an acceptable trade-off value. Thisyields 41% efficiency which will again improve once 𝑤

1is

optimized.

(5) Optimizing Outer Diameter and Line Width of PrimaryPSC (𝑑

𝑜1, 𝑤1). In this step, the primary PSC is visited back.

The width of the conductor 𝑤1is increased from 𝑤

1min to itsoptimal value and the outer diameter 𝑑

𝑜1is increased from its

original value. Increasing 𝑤1comes with decreasing the coil

resistance 𝑅dc as defined by (4). This leads to higher qualityfactor𝑄

1of the primary coil and results in better 𝜂

12efficiency

values. However, increasing 𝑤1also requires larger coils with

higher 𝑑𝑜1

values. Observing 𝜂12

curves for various 𝑑𝑜1

and𝑤1has shown that the highest efficiency values are guaranteed

for 𝑑𝑜1

> 75mm. Considering 𝑑𝑜1= 75mm along with 𝑑

𝑖1=

8mm leads to 𝜑1= 0.8. Using these settings for 𝑑

𝑜1and 𝜑

1,

𝜂12is calculated and reported versus increasing𝑤

1(Figure 6).

As shown in Figure 6, 𝜂12

improves with increasing 𝑤1and

moves towards a saturation value around 𝑤1

= 3mm. Tominimize the mutual coupling between conductors of thesame coil [24], we select 𝑤

1= 2.2mm as an optimal value

with almost 75% efficiency achieved.In comparison with Figure 4 (Step 3), we are achieving in

this step an efficiency improvement of about 45%.

Table 2: Optimal inductive link coil designs by using IterativeProcedure.

Parameters Primarycoil

Secondarycoil

Outer diameter (𝑑𝑜) 80 (mm) 20 (mm)

Inner diameter (𝑑𝑖) 10 (mm) 11 (mm)

Width of conductor (𝑤) 2.5 (mm) 0.3 (mm)Spacing between conductors (𝑠) 7.5 (mm) 0.6 (mm)Number of turns (𝑛) 4 5𝜂12at 𝑓 = 13.56MHz 78%

40

50

60

70

80

90

0 1 2 3 5Effi

cien

cy (%

)

w1 (mm)

Figure 6: Calculated efficiency (𝜂12) versus (𝑤

1) for Step 5 optimiza-

tion.

(6) Iteration by Going to Step 3, Optimized Design, andEfficiency Results. Efficiency values from Step 5 significantlyimprove in comparison to Step 3. However, we can achievefurther improvement by iterating through Steps 3–5. Iter-ations can continue until less than 0.1% improvement isobtained per iteration. Finally, the optimal link parametersrealized using the IP approach are summarized in Table 2.For this optimized geometry, the link efficiency is enhancedto 78%. However, during the implementation of the currentIP procedure, we faced a difficulty in defining the best valuefor conductor widths (𝑤

1and 𝑤

2) while simultaneously

optimizing the space between conductors (𝑠1and 𝑠2). This

will be quite possible by using the Genetic Algorithms (GA)based approach that will allow the optimization of both s andw parameters at the same time as it will be shown in the nextsection.

3.2. Genetic Algorithms (GA) Based Approach. Genetic Algo-rithms (GA) are population based search metaheuristicscapable of generating solutions for problem optimizationsusing mechanisms inspired from the natural evolutionprocess. GA uses a population with different individualscarrying chromosomes that represent the parameters tobe optimized. Chromosomes are changed in a way sim-ilar to gene evolution, mainly through three operators:selection, crossover, and mutation. Chromosomes with bet-ter performances have more chances to survive into thenext generation. Repeating these steps, the individuals inthe population will become better and better, eventually


reaching the optimum. The construction of a GA optimiza-tion for the studied implant is conducted along a workflowdivided into five major parts as follows.

(1) System Specification. This consists in setting first thedesign constraints as enforced by the implant system. Theseconstraints are those defined in Table 1. In addition, welimited the size of the primary PSC with using 𝑑

𝑜1= 80mm

and choose 𝑑𝑖2

= 11mm to leave some space for a chipto be placed at the center of the secondary PSC. After that,the fitness or objective function is defined along with theparameters to be optimized. For our case, 𝑤

1, 𝑤2, 𝑠1, and 𝑠

2

are selected as the optimization variables and the maximumpower efficiency is our optimization target. The objectiveis to optimize the geometries of the coils, while the targetefficiency is calculated using the electric parameters definedby (1) to (8).

(2) Initialize Population. In this step, all individuals are ran-domly generated to initialize the population with respectingupper and lower limits of the different coil parameters. Theappropriate population size is carefully chosen as higheraccuracy of GA is obtained for increasing population sizes.However, excessive increase in population sizes can signifi-cantly slow down the speed of the system to converge.

(3) Evaluate the Fitness of EachChromosome in the Population.The fitness of all individuals in the population is calculated,so that best individuals are selected in a selection step. In ourcase, the fitness function accepts a vector of the geometricvariables𝑤

1,𝑤2, 𝑠1, and 𝑠

2to be optimized.These variables are

encoded into a bit stream to be used just like a chromosome.Each bit stream represents an individual and is a combinationof all optimized parameters. For each individual of thepopulation, the actual design parameters are plugged into (1)to (8) to derive the link efficiency 𝜂

12that is a measure of the

fitness value for that individual. To avoid also unreasonabledesign solutions, upper and lower limits are assigned for thedifferent candidate design solutions.

(4) Selection, Crossover, and Mutation. Appropriate selec-tion, crossover, and mutation operators are implemented.Selection is a key operation in GA. It aims to select betterindividuals in a population. We used the Roulette wheelselection for this operator. It is a selection method basedon proportional probability as presented in [25]. Some ofthe selected chromosomes are recombined and their genesexchanged through a crossover operation.The crossover cre-ates better individuals by recombining some of the selectedchromosomes. A crossover probability 𝑃

𝑐is used to deter-

mine whether an individual is going to do crossover withothers or not [26]. In our study, the probability𝑃

𝑐is randomly

selected within [𝑃𝑐max, 𝑃𝑐min] range similarly as used in [27].

To avoid premature convergence,𝑃𝑐max and𝑃

𝑐min are carefullyselected to 0.9 and 0.8, respectively. Finally, a mutationoperation is necessary to create new individuals to get GAjump out of local optimums. This is implemented using aprobability𝑃

𝑚for a population to create new individuals. Low

𝑃𝑚values may cause GA to fall rapidly into local optimums.

Table 3: Optimal inductive link coil designs by using geneticalgorithms.


Secondarycoil



Conductors’ width (𝑤) 1.5 (mm) 0.25 (mm)Spacing between conductors (𝑠) 3 (mm) 0.35 (mm)Number of turns (𝑛) 8 8𝜂12at 𝑓 = 13.56MHz 80.6%

However, excessive 𝑃𝑚values will slow down the converging

speed of themethod. 0.001–0.01 is a commonly used range for𝑃𝑚as presented in [28, 29].

(5) Meet Convergence Stopping Criteria. The algorithm endseither when amaximumnumber of generations are producedorwhen a satisfactory optimization level is reached.When thestopping criteria are met, the GA process is completed andthe optimal efficiency value is extracted. Otherwise, a newpopulation is generated as defined by the GA rules and Step3 is restarted.

For the current design case, a MATLAB software toolimplementing Genetic Algorithms is used. To run this tool,an M-file that computes the objective function to be opti-mized is first developed. Carrying out the GA optimizationhas achieved the optimal geometric parameters for theactual implant coils. The obtained optimal parameters aresummarized in Table 3. Using these geometries in (1) to (8),80.6% efficiency is obtained.This is a slightly better efficiencyvalue in comparison with what we get using the IP approach.However, if we consider the implementation burden of IP,the GA based algorithms help perform the optimization in amore systematic and efficient way. In addition, GA took veryfew cycles to converge and once properly configured, it takesvery little time to deliver an adequate design.

4. GA-FEM Numeric Optimization Approach

This section discusses a purely numeric GA-FEM optimiza-tion approach. This approach is implemented using a par-ticular software module that is integrated within HFSS FEMsimulator [18]. It consists in coupling finite element numericsimulation method with GA based search strategy. Simi-lar techniques have been previously proposed to optimizeantennas and waveguide systems [30–32]. In these previousstudies, FEM simulators are connected with MATLAB tosolve the related problems. For the present case, GeneticAlgorithmoptimization is already integratedwithHFSS FEMsimulator. Implementing GA-FEM consists in three mainphases. For each phase, particular modeling and design tasksare required. First, initial geometry parameters with specificdesign constraints are inserted to restrict the search regionand prevent the optimizer from creating physically meaning-less designs (such as overlapping and overtaking of the geom-etry). The second phase runs the “GA-FEM-optimizer” that


0 40 80

(mm)

Figure 7: 3D model of initial geometry defined for GA-FEM opti-mization.

Table 4: Optimal inductive link coil designs by using GA-FEM-optimizer.


Secondarycoil



Conductors’ width (𝑤) 1.9 (mm) 0.7 (mm)Spacing between conductors (𝑠) 2.5 (mm) 0.52 (mm)Number of turns (𝑛) 8 4𝜂12at 𝑓 = 13.56MHz 78%

implements a Genetic Algorithm based search to produce anew structure starting from the initial geometry settings. Inthis phase, a fitness or objective value function is computedand appropriate selection, crossover, and mutation operatorsare used.

For the current design case, the initial geometry struc-tures for the primary and secondary coils are HFSS modeledas shown in Figure 7. The variables selected to be optimizedare 𝑤

1, 𝑤2, 𝑠1, and 𝑠

2and their lower and upper limits

are fixed. The maximum power efficiency is again ouroptimization target and the fitness or objective function isdefined accordingly. GA-FEM also optimizes the geometriesof the coils. However, in this case, the electric parameters suchas resistances, inductances, coupling constants, and qualityfactors of the coils are obtained using numerical simulations.The simulated electric parameters are used again with (8)to evaluate the maximum efficiency and measure the fitnessvalues of all individuals for a given generation. To move fromone generation to the next, the same settings we used for GAare here implemented for selection, crossover, and mutationoperators with the same rate of mutation and crossoverrecombination. 90% target efficiency objective is fixed and 60hours maximum simulation time is defined for the systemto converge to the desired goal. The simulation ends eitherif the desired optimization level is met or if the maximumsimulation time is reached.

Running GA-FEM numeric optimization, we were notable to achieve the 90% desired target. However, we ended

0102030405060708090

1 4 7 10 13 16 19

Ref [19]PI

GAGA-FEM-optimizer

Effici

ency

(%)

Frequency (MHz)

Figure 8: Simulated efficiency values for GA-FEM, GA, and IPdesign optimizations compared to [19].

after 60 hours of simulation time with the optimized geome-tries which is illustrated in Table 4

The simulated efficiency values of the optimized coils withGA-FEM are given in Figure 8 together with the efficiencyresults for the PI, GA, and [19] optimized designs. Theseefficiency values are reported versus frequency as they arebased on the coils electric parameters that are simulatedfor [0–20MHz] frequency range. Particularly, for 13.56MHzoperating frequency, GA-FEM is capable of achieving 78%efficiency as reported in Table 4. In reference to GA-FEM, theefficiency losses of the approximate analytic approaches areevaluated. As reported in Figure 8, 9.7% and 5.7% efficiencydegradation are obtained at the operating frequency for IPand GA, respectively. However, more than 16% efficiencylosses are observed when comparing GA-FEM to [19]. Thisoutlines the outperformance of the proposed analytical IPand GA methods in comparison with [19].

Compared to IP and GA analytical approaches, theoutperformance of GA-FEM is clearly illustrated in Figure 8.However, if we consider the long simulation time of GA-FEM and also the modeling and design efforts required toimplement it, one can easily conclude that GA, although it isrelying on approximate analytic circuit modeling, is capableof competing with rapid and systematic design optimizationsand with comparable performances. In addition, the GAoptimized design can also serve to define the initial designstructure for GA-FEM optimization to get further perfor-mance improvement with augmented convergence speed.Actually, when GA is used to provide the initial parametersfor GA-FEM (GA-FEM-initialized), the same efficiency ofthe previous GA-FEM optimal design (78%) is obtained afteronly 3 h : 52 mn of computing time in comparison with 60hours previously needed.However, if we leave the simulationscomplete to the end, the systemwill converge to a design witheven better efficiency values as shown in Figure 9. In this fig-ure, around 81% efficiency is observed at 13.56MHz operatingfrequency for GA-FEM-initialized with GA. Actually, GA-FEM-initialized achieves an improved performance in com-parison with GA-FEM. Moreover, noticeable convergence


0

12.5

25

37.5

50

62.5

75

87.5

1 4 7 10 13 16 19

Ref [19]PIGA

GA-FEM-optimizerGA-FEM-optimizer

Effici

ency

(%)

Frequency (MHz)

Figure 9: Simulated efficiency values for GA-FEM-initialized incomparison with GA-FEM, GA, IP, and [19].

speed acceleration is observed which represents an addi-tional outcome for the proposed GA analytical basedoptimization.

5. Experimental Measurement Results

To confirm the validity of the proposed approaches, experi-mental measurements are required. For this aim, the differ-ent designed coils are fabricated on printed circuit boards(PCB) using 0.035mm copper and FR4 Epoxy substrates[8, 22]. The first fabricated PCB is related to the primaryand secondary coils of reference design [19]. The secondand third designed coils are fabricated according to theoptimal geometries designed, respectively, with IP and GAapproaches. The last PCB, which is shown as an examplein Figure 10, is fabricated as optimized using the GA-FEMapproach.

For all the PCB fabricated coils, the system parameters aremeasured using a network analyzer reference VNA MasterAnritsu MS2036A. This network analyzer was calibratedsuch that all cable influences are removed. The experimentalsystem used to measure the efficiency of the fabricated coilsis presented in Figure 11. This figure is showing particulararrangements made to ensure coil alignments with 30mmseparation distance between the primary and secondarycoils.

For each design case, the network analyzer is used toperform two-port measurements from 1MHz to 20MHz oneach pair of coils. To extract the experimental efficiencyvalues, 𝑆-parameters are measured and then converted into𝑍-parameters using a dedicatedMATLAB script [33, 34].The

Table 5: Analytical, numerical, and experimental results for alloptimization approaches.

Approaches Analyticalresults

Numericalresults

Experimentalresults

According to [19] 65.5% 67.7% 63.4%Iterative approach 78.6 % 70.4% 68%Genetic algorithms 80.6% 73.5% 72%GA-FEM-optimizer 78% 77.9%

quality factor of each coil is determined using the followingequations:

𝑄1=Im (𝑍

11)

Re (𝑍11);

𝑄2=Im (𝑍

22)

Re (𝑍22).

(10)

The coupling factor between the primary and secondary coilis expressed as

𝑘12

= √Im (𝑍

12) Im (𝑍

21)

Im (𝑍22) Im (𝑍

11), (11)

where 𝑄1, 𝑄2, and 𝑘

12are substituted in (8) to get the effi-

ciency values. The obtained efficiency values for the differentdesign cases are shown in Figure 12.

Referring to Figure 12, the efficiency values at 13.56MHzoperating frequency are extracted and summarized in Table 5together with the analytic and simulated results for the differ-ent approaches.This table clearly shows that the experimentalresults are in close agreement with what has been analyticallyand numerically predicted. Slight differences are observedbetween measured and simulated efficiency results. Compar-ing the different methods, the accuracy of GA-FEM is clearwith only very small difference (0.1% at 13.56MHz) betweenmeasurements and simulations. This high accuracy can beexplained by the fact that purely numerical optimization isimplemented taking into account real resistances, parasiticcapacitances, and all kinds of coupling effects between the coiland its environment.

Apart fromGA-FEM accuracy, it is important to compareand validate the accuracy of the proposed IP and GAanalytical approaches. Comparing analytical to experimentalresults from Table 5, 16% errors are obtained for IP and only12% errors are shown for GA.These are reasonable errors forboth IP and GA if we consider the simplified mathematicalmodels we used to approximate the coils electric parameters.These analytical models are derived with ignoring all types oflosses such as capacitive, proximity effect and eddy currentlosses. Such losses make the actual coil resistances largerthan those analytically predicted. Larger real resistance valuesgenerate lower coil quality factors and thus lower measuredefficiencies are obtained in comparisonwith analytical values.

Comparing GA to IP, GA is more accurate; this is dueto higher accurate inductance values predicted for coilsoptimized with GA. Actually, it is demonstrated according


(a) (b)

Figure 10: Photograph of the fabricated printed spiral coil optimized using GA-FEM: (a) primary coil and (b) secondary coil.

Figure 11: Experimental setup for measuring link efficiency using anetwork analyzer.

to [35] that the accuracy of the inductance model defined by(1) decreases for increasing (𝑠/𝑤) ratios. Thus, if we comparethe coil geometries optimized with IP and GA, it is clear fromTables 2 and 3 that 𝑠

1/𝑤1= 3 and 𝑠

2/𝑤2= 2 for the IP design

versus 𝑠1/𝑤1= 2 and 𝑠

2/𝑤2= 1.34 for GA. Thus, inductance

values derived from (1) for both primary and secondary coilsare more accurate for GA as studied in [35]. Actually, withGA, we were able to optimize both s- and w-parameters atthe same time contrary to IP. This results in higher accuracyof GA in comparison with IP.

At the end, higher efficiency values are experimented forGA-FEM (77.9% at 13.56MHz) in comparison with GA (72%at 13.56MHz). Using GA-FEM as a reference design, GA iscapable of delivering an accurate enough design with around7% experimental performance degradation.This is acceptablegiven the approximate modeling used for GA. However, GA-FEM requires a long simulation time to converge and onecan use GA to initialize GA-FEM and thus considerably

0

10

20

30

40

50

60

70

80

90

1 7 13 20

Ref [19]PI

GAGA-FEM-optimizer

Effici

ency

(%)

Frequency (MHz)

X: 13.57Y: 77.81

X: 13.57Y: 63.31

Figure 12: Measured efficiency values for GA-FEM, GA, and IPdesign optimizations compared to reference design [19].

accelerate its convergence speed with further performanceimprovements.

6. Conclusions

This paper aims for geometry optimization of inductivelycoupled printed spiral coils (PSC) for powering a givenimplantable sensor system. This is a complex nonlinearoptimization problem with many interfering design param-eters. Our objective is to implement and compare differentoptimization approaches capable of achieving, in a simpleand a systematic way, optimal and accurate link designswithin acceptable design times.

In this context, analytic approaches are first proposed.These approaches are based on simple mathematical modelsthat approximate the electric parameters of the coils and the


power transfer efficiency. Using these models, an IterativeProcedure (IP) is first developed. IP is a traditional popularapproach for optimizing coil geometries. Suchmethod suffersfrom ambiguity, heaviness, and complexity shortcomings.However, improved performance results are obtained usingthe proposed IP approach in comparison with a referencedesign for the same system. Advanced population basedsearch strategies implementing Genetic Algorithms (GA) arethen used. Actually, it was shown that GA is easier to imple-ment and is capable of generating better optimized designsin a more systematic and efficient way. GA is also capableof delivering the adequate design within short computationtimes.

Both IP and GA analytical approaches are compared toGA-FEM, a purely simulation based optimization method.GA-FEM consists in coupling finite element simulations withGA based search. Performed simulation results confirmedthat GA-FEM achieves, at the resonant frequency, the highestperformance. In comparison with GA-FEM, acceptable 7%efficiency losses are observed for GA. Despite its highestperformance, GA-FEM suffers from extensive processing andvery long design time. To minimize its processing time andfurther improve its performance, GA-FEM can be initializedusing GA. In this case, even better performance results areobtained with higher convergence speed.

Experimental validation has been performed for all thedesigned cases. For this aim, the different coils optimizedwith the different approaches are fabricated on printed circuitboards (PCB).The system parameters are thenmeasured andthe transfer power efficiencies are determined. It is shown thatthe proposed IP and GA analytical approaches are accurateenough despite the approximatedmathematical models used.Comparing GA to IP, GA is more accurate as we were ableto better control particular design parameters. This leads tohigher accurate inductances predicted for coils optimizedwith GA. Finally, the highest performance and accuracyof GA-FEM is confirmed with only minor errors observedbetween measurements and simulation results.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This project was funded by the King Abdulaziz City forScience and Technology (KACST), under Grant no. 91-34- ��

��. The research team acknowledges and expresses their

gratitude to KACST for the technical and financial support.

References

[1] A. Trigui, S. Mehri, A. C. Ammari, J. Ben Hadj Slama, and M.Sawan, “Prosthetic power supplies,” in Wiley Encyclopedia ofElectrical and Electronics Engineering, John Wiley & Sons, NewYork, NY, USA, 2015.

[2] T. Sun, X. Xie, and Z. Wang, Wirless Power Management,Springer, New York, NY, USA, 2013.

[3] K. Van Schuylenbergh and R. Puers, The Concept of InductivePowring, Springer Science+Business Media BV, Dordrecht,Netherlands, 2009.

[4] G. A. Kendir, W. Liu, G. Wang et al., “An optimal designmethodology for inductive power link with class-E amplifier,”IEEE Transactions on Circuits and Systems I: Regular Papers, vol.52, no. 5, pp. 857–866, 2005.

[5] B. Lenaerts and R. Puers, “Inductive link design,” in Omnidi-rectional Inductive Powering for Biomedical Implants, AnalogCircuits and Signal Processing, pp. 39–81, Springer, Dordrecht,The Netherlands, 2009.

[6] A. Trigui, S.Hached, F.Mounaim,A.C.Ammari, andM. Sawan,“Inductive power transfer system with self-calibrated primaryresonant frequency,” IEEE Transactions on Power Electronics,vol. 30, no. 11, pp. 6078–6087, 2015.

[7] M. Kiani and M. Ghovanloo, “A figure-of-merit for designinghigh-performance inductive power transmission links,” IEEETransactions on Industrial Electronics, vol. 60, no. 11, pp. 5292–5305, 2013.

[8] U.-M. Jow and M. Ghovanloo, “Design and optimization ofprinted spiral coils for efficient transcutaneous inductive powertransmission,” IEEE Transactions on Biomedical Circuits andSystems, vol. 1, no. 3, pp. 193–202, 2007.

[9] H. Peng, J. Zhao, H. Zhang et al., “A novel approach to designof microstrip UWB bandpass filter using modified geneticalgorithm,” Progress In Electromagnetics ResearchM, vol. 36, pp.169–175, 2014.

[10] J. T. Rayno, J. S. K. Nakatsu, G. C. Huang, N. Celik, and M. F.Iskander, “3D metamaterial broadband ground plane designedusing genetic programming for the long slot array antenna,”in Proceedings of the IEEE Antennas and Propagation SocietyInternational Symposium (APSURSI ’13), pp. 400–401, Orlando,Fla, USA, July 2013.

[11] R. Bera, D. Mandal, R. Kar, and S. P. Ghoshal, “Application ofimproved Particle Swarm Optimization technique for thinningof concentric hexagonal array antenna,” in Proceedings of the4th World Congress on IEEE Information and CommunicationTechnologies (WICT ’14), pp. 188–193, Bandar Hilir, Malaysia,December 2014.

[12] M. Angelova and T. Pencheva, “Tuning genetic algorithmparameters to improve convergence time,” International Journalof Chemical Engineering, vol. 2011, Article ID 646917, 7 pages,2011.

[13] P. P. Chandran and S. Viswasom, “Gain and bandwidth opti-mization of a novel microstrip patch antenna,” in Proceedings ofthe 4th International Conference on Advances in Computing andCommunications (ICACC ’14), pp. 315–318, IEEE, Cochin, India,August 2014.

[14] M. Ural and C. Bayseferogullari, “Solution of the antenna place-ment problem by means of global optimization techniques,” inProceedings of the 20th International Conference onMicrowaves,Radar and Wireless Communications (MIKON ’14), pp. 1–4,Gdansk, Poland, June 2014.

[15] A. Pichler, D. Steffelbauer, and A. Nazarov, “Examples forgenetic algorithm based optimal RFID tag antenna design,”in Proceedings of the IEEE RFID Technology and ApplicationsConference (RFID-TA ’14), pp. 223–227, IEEE, Tampere, Finland,September 2014.

[16] S. Song and R. D. Murch, “An efficient approach for optimizingfrequency reconfigurable pixel antennas using genetic algo-rithms,” IEEE Transactions on Antennas and Propagation, vol.62, no. 2, pp. 609–620, 2014.


[17] S. Mehri, J. Ben Hadj Slama, A. C. Ammari, and H. Rmili,“Genetic algorithm based geometry optimization of inductivelycoupled printed spiral coils for remote powering of electronicimplantable devices,” in Proceedings of the Global Summit onComputer and Information Technology (GSCIT ’14), pp. 1–6,Sousse, Tunisia, June 2014.

[18] HFSS Online Help, April 2015, http://www.ansys.com/.[19] E. G. Kilinc, C. Dehollain, and F. Maloberti, “Design and

optimization of inductive power transmission for implantablesensor system,” inProceedings of the 11th InternationalWorkshopon Symbolic and Numerical Methods, Modeling and Applicationsto Circuit Design (SM2ACD ’10), pp. 1–5, IEEE, Gammath,Tunisia, October 2010.

[20] A. K. RamRakhyani, S. Mirabbasi, and M. Chiao, “Designand optimization of resonance-based efficient wireless powerdelivery systems for biomedical implants,” IEEETransactions onBiomedical Circuits and Systems, vol. 5, no. 1, pp. 48–63, 2011.

[21] U.-M. Jow and M. Ghovanloo, “Modeling and optimization ofprinted spiral coils in air, saline, and muscle tissue environ-ments,” IEEE Transactions on Biomedical Circuits and Systems,vol. 3, no. 5, pp. 339–347, 2009.

[22] M. Zargham and P. G. Gulak, “Maximum achievable efficiencyin near-field coupled power-transfer systems,” IEEE Transac-tions on Biomedical Circuits and Systems, vol. 6, no. 3, pp. 228–245, 2012.

[23] W. B. Kuhn and N. M. Ibrahim, “Analysis of current crowdingeffects in multiturn spiral inductors,” IEEE Transactions onMicrowaveTheory and Techniques, vol. 49, no. 1, pp. 31–38, 2001.

[24] H. Kondori, M. A. Mansouri-Birjandi, and S. Tavakoli, “Reduc-ing mutual coupling in microstrip array antenna using meta-material spiral resonator,” International Journal of ComputerScience Issues, vol. 9, no. 3(1), pp. 52–56, 2012.

[25] L. Zhang, H. Chang, and R. Xu, “Equal-width partitioningroulette wheel selection in genetic algorithm,” in Proceedingsof the Conference on Technologies and Applications of Artifi-cial Intelligence (TAAI ’12), pp. 62–67, IEEE, Tainan, Taiwan,November 2012.

[26] A. Sharma and M. Sinha, “Influence of crossover and mutationon the behavior of Genetic algorithms in mobile Ad-hocnetworks,” in Proceedings of the 8th International Conferenceon Computing for Sustainable Global Development (INDIACom’14), pp. 895–899, New Delhi, India, March 2014.

[27] C. Pan, S. Cao, K. Zheng, and Z. He, “Optimizing appropriatetwo probable value method by adaptive genetic algorithm,” inProceedings of the International Conference on ComputationalProblem-Solving (ICCP ’12), pp. 171–173, Leshan, China,October2012.

[28] T.D.Nguyen, T. P. Vuong, Y.Duroc, andV. Y. Vu, “Optimizationof pifa antenna using an auto-embedded genetic algorithm,” inProceedings of the 3rd International Conference on Communica-tions and Electronics (ICCE ’10), pp. 367–372, IEEE, Nha Trang,Vietnam, August 2010.

[29] Y. He, J. Ao, X. Tang, and J. Yang, “The optimum design ofPIFA based on HFSS and genetic algorithm,” in Proceedings ofthe 7th International Conference on Wireless Communications,Networking and Mobile Computing (WiCOM ’11), pp. 1–4,Wuhan, China, September 2011.

[30] W.-W. Lee, S.-Y. Chen, T.-C. Lin, H.-T. Chou, and H.-T. Hsu,“Integration of HFSS and genetic algorithm for the optimumdesign of waveguide components,” in Proceedings of the IEEEAntennas and Propagation Society International Symposium

(AP-S ’07), pp. 2225–2228, IEEE, Honolulu, Hawaii, USA, June2007.

[31] S. Shu-Hui andW. Bing-Zhong, “Parameter optimization basedon GA and HFSS,” Journal of Electronic Science and Technologyof China, vol. 3, no. 1, pp. 45–47, 2005.

[32] A. J. Kerkhoff and H. Ling, “Design of a band-notched planarmonopole antenna using genetic algorithm optimization,” IEEETransactions on Antennas and Propagation, vol. 55, no. 3, pp.604–610, 2007.

[33] D. Pozar,Microwave Engineering, JohnWiley& Sons, NewYork,NY, USA, 2nd edition, 1998.

[34] M. Kiani, U.-M. Jow, and M. Ghovanloo, “Design and opti-mization of a 3-coil inductive link for efficient wireless powertransmission,” IEEE Transactions on Biomedical Circuits andSystems, vol. 5, no. 6, pp. 579–591, 2011.

[35] N. Troedsson and H. Sjoland, “Monolithic inductor modelingand optimization,” in Radio Design in Nanometer Technologies,pp. 217–240, Springer, Berlin, Germany, 2006.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Documents

Research Article Geometry Optimization Approaches of …downloads.hindawi.com/journals/js/2016/4869571.pdf · is usually very small. is paper is dealing with geometry optimization